192,225 research outputs found
Subsystems of Transitive Subshifts with Linear Complexity
We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380]
On lattice profile of the elliptic curve linear congruential generators
Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG
Average Case Tractability of Non-homogeneous Tensor Product Problems
We study d-variate approximation problems in the average case setting with
respect to a zero-mean Gaussian measure. Our interest is focused on measures
having a structure of non-homogeneous linear tensor product, where covariance
kernel is a product of univariate kernels. We consider the normalized average
error of algorithms that use finitely many evaluations of arbitrary linear
functionals. The information complexity is defined as the minimal number n(h,d)
of such evaluations for error in the d-variate case to be at most h. The growth
of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the
covariance operator and determines whether a problem is tractable or not. Four
types of tractability are studied and for each of them we find the necessary
and sufficient conditions in terms of the eigenvalues of univariate kernels. We
illustrate our results by considering approximation problems related to the
product of Korobov kernels characterized by a weights g_k and smoothnesses r_k.
We assume that weights are non-increasing and smoothness parameters are
non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for
some non-increasing function g. In particular, we show that approximation
problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d
and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k
>1. For other types of tractability we also show necessary and sufficient
conditions in terms of the sequences g_k and r_k
Maximum-order Complexity and Correlation Measures
We estimate the maximum-order complexity of a binary sequence in terms of its
correlation measures. Roughly speaking, we show that any sequence with small
correlation measure up to a sufficiently large order cannot have very small
maximum-order complexity
Error linear complexity measures for multisequences
Complexity measures for sequences over finite fields, such as the linear complexity and the k-error linear complexity, play an important role in cryptology. Recent developments in stream ciphers point towards an interest in word-based stream ciphers, which require the study of the complexity of multisequences. We introduce various options for error linear complexity measures for multisequences. For finite multisequences as well as for periodic multisequences with prime period, we present formulas for the number of multisequences with given error linear complexity for several cases, and we present lower bounds for the expected error linear complexity
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